STING DYNAMICS OF WIND TUNNEL MODELS

, ,VON KARMAN GAS DYNAMICS FACILITY

ARNOLD ENGINEERING DEVELOPMENT CENTERAIR FORCE SYSTEMS COMMAND

ARNOLD AIR FORCE STATION, TENNESSEE 37389

May 1976

Final Report for Period July 1, 1974 - June 30, 1975

Approved for public release; distribution unlimited.

Prepared for

DIRECTORATE OF TECHNOLOGY (DY)ARNOLD ENGINEERING DEVELOPMENT CENTERARNOLD AIR FORCE STATION, TENNESSEE 37389

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This technical report has been reviewed and is approved for publication.

FOR THE COMMANDER

KENNETH B. HARWOODMajor, CFResearch & Development

DivisionDirectorate of Technology

ROBERT O. DIETZDirector of Technology

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AEDC-TR-76-4l4. TITLE (and Subtitle) S. TYPE OF REPORT 1\ PERIOD COVERED

Final Report - July 1,STING DYNAMICS OF WIND TUNNEL MODELS 1974 - June 30, 1975

6. PERFORMING ORG. REPORT NUMBER

7. AUTHOR(.) 8. CONTRACT OR GRANT NUMBER(.)

James P. Billingsley - ARO, Inc.9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT, TASK

Arnold Engineering Development Center (DY) AREA 1\ WORK UNIT NUMBERSAir Force Systems Command Program Element 65807FArnold Air Force Station, Tennessee 37389

II. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE

Arnold Engineering Development Center (DYFS) May 1976Air Force Systems Command 13. NUMBER OF PAGESArnold Air Force Station, Tennessee 37389 6214. MONITORING AGENCY NAME 1\ ADDRESS(II dillerent Irom Con troll/nil Ollice) IS. SECURITY CLASS. (01 this report)

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wind tunnel models mechanical vibrationdynamic response mathematical analysisstructural response data acquisitionvibration damping accuracy

20. ABSTRACT (Continue on reverse side If necessary and Identify by block number)

Wind tunnel model support stings can be subjected to transientaerodynamic and inertial loads which will create oscillatory trans-lations and angular deflections. These transient oscillationsalways impair steady-state data accuracy and can be large enough tocause structural failure. The primary result of this investigationhas been the formulation of a mathematical analysis for the dynamicresponse of sting-balance combinations subjected to arbitrary

DO FORM1 JAN 73 1473 EDITION OF 1 NOV 6S IS OBSOLETE

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20. ABSTRACT (Continued)

transient load inputs. The analysis also provides for sting rigidbody motion so that model to tunnel injection and continuous stingrotation (a sweep) can be simulated. A computer program waswritten to numerically solve the sting-model-balance notion equa-tions. A necessary and supplemental result to the basic analysiswas the development of a computer program to compute deflections,first natural frequency, and spring constants for cantilever beamswith mixed tapered and untapered section of different materials.Two primary applications of these programs and the associatedanalysis are given.

AFSCArnold AFS Tenn

UNCLASSIFIED

AEDC-TR-76-41

PREFACE

The work reported herein was conducted by the Arnold EngineeringDevelopment Center (AEDC), Air Force Systems Command (AFSC),under Program Element 65807F. The results were obtained by ARO,Inc., (a subsidiary of Sverdrup & Parcel and Associates, Inc.), contractoperator of AEDC, AFSC, Arnold Air Force Station, Tennessee. Thisresearch was done under Project No. V37A-32A in support of the HighReynolds Number Tunnel (HIRT) Project. The author of this reportwas James P. Billingsley, ARO, Inc. The manuscript (ARO controlNo. ARO-VKF-TR-75-150) was submitted for publication on October13, 1975.

The author gratefully acknowledges the comments, suggestions,and technical information he has received from the VKF HIRT andTunnel F group personnel. . Special thanks are extended to Mr. JackCoats l Tunnel F Project Engineer, for supplying the detailed technicalinformation required for the Tunnel F application of the present analysis.

1

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CONTENTS

2.32.42.5

1.02.0

3.0

4.0

INTRODUCTION • . . .ANALYSIS2. 1 Sting Materials. • . . • . . • • .2.2 Sting Deflection, Stress. Strain, and Natural

Frequency . . . . . . . . . . . . . . .Influence Coefficients (Spring Constants)Sting Natural Frequencies and Mode Shapes .•Lagrangian Formulation of Sting- Model MotionEquations . . ..

APPLICATIONS3.1 HIRT Sting Design and Transient Response3.2 VKF Tunnel F Sting Dynamics .SUMMARY4. 1 Sting Structural Design • •4.2 Sting Structural Dynamics • . • • •REFERENCES. • • . . . . . • . • • .

ILLUSTRATIONS

Page

7

9

111518

18

3540

505152

Figure

1. Simple Sting Deflection Geometry . • • . •

2. Beam Geometry and Basic Input for STING-1

7

Program . . . . . . . . . . . . . . . .

3. Maximum Static Bending Stress in the Sting .

4. Lowest Natural Frequency of the Example Stingfor Various Model Weights and Sting Materials

5. Basic Bending Mode Shapes for the ExampleCarbide-Steel Sting... • 0 • • • • • • • • . . .

11

13

14

19

6. Sting Coordinate Systemsa. Planar Deflection Geometry of the Sting-Model

Concentrated Mass Representation • • • • •b. Coordinates 6f Mass Element with Respect to

Moving Axis Systems • • • • • . . . . • •c. Coordinates of Mass Element with Respect to

an Inertial Axis System • • • • • • • . . •

3

. .

. .20

22

23

AEDC·TR·76·41

Figure

d. Final Coordinates of Deflected Sting withRespect of Undeflected Sting . • • .

7. Axial Force Bending Moment Schematic • . .

24

29

8. Predicted Variation of Dynamic Pressure Ratio in aPilot Ludweig-Tube Wind Tunnel . . . . • . . • .. 35

9. Motion History for ATT Model and Steel Sting,Condition No.7, Basic qoo Variationa. Cl'p ::; O. 60 deg. . . . . . . • • . . • . . 36b. Cl'p::; 0.25, 0.50, and 0.75 deg . . . . . . 37

10. Motion History for ATT Model and Carbide-SteelSting, Condition No.7, Basic qoo Variation, Cl'p ::;5.0deg. . . . . . . . . . . . . . . . . . . . . . 39

11. Motion History for ATT Model and Carbide-SteelSting, Condition No.7, qoo ::; qoo ,Cl'p ::; 5.0 deg. 39

max12. VKF Hypervelocity Tunnel F Stings

a. Actual Sting Configuration. . . . . . . . . • •. 41b. Structural Stiffness Synthesis . . ',' . . . 41

13. Transient Aerodynamic Loads on the Tunnel FMo.del . . . . . . . . . . . . . . . .

14. Transient Flow Velocity History for VKFTunnel F . . . . . . . . . . . . . . .

42

42

50

15. Theoretical and Experimental Angular Deflectionsof the Tunnel F Modela. Transient Amplitude without Structural Damping.. 45b. Transient Amplitude with Structural Damping 45c. Average Amplitude . . • . . . . . • . . . . 46d. Oscillation Amplitude . . . • • . . . . . . . 46

16. Theoretical and Experimental Translational ModelNose Tip Deflectionsa. Transient Amplitude without Structural Damping 47b. Transient Amplitude with Structural Damping 47c. Average Amplitude . . . . . . • • . . . . • 48d. Oscillation Amplitude . • . . . . • . • . . . 48

17. Model Nose Tip Translational Deflection Geometry .. 49

18. Theoretical Results for Cl'plunge and Model cgTranslation (z 1)' • . . . . • • • . • . . .

4

TABLES

1. Characteristics of Sting Construction Materials.

2. HIRT Sting Deflections and Natural Frequencies

AEDC·TR·76·41

Page

10

14

NOMENCLATURE. . . . . . . . . . . . . . . . .

5

55

AEDC-TR-76-41

1.0 INTRODUCTION

It is highly desirable to more accurately measure "second-order"aerodynamic effects resulting, for example, from Reynolds number in-fluence. Consequently, all wind tunnel data systems are required toprovide a high degree of accuracy. One area that requires thoroughanalysis is the magnitude of sting bending and oscillations that will existunder high free-stream dynamic pressure and impulse starting loadscharacteristic of some wind tunnels (Ref, 1).

The majority of aerodynamic data available from wind tunnels isobtained from captive (restrained) model testing. In this situation, themodel is supported at the free end of a beam (the sting) which is canti-levered forward from a support sector into the oncoming airflow in thetest section. Aerodynamic forces and moments are typically measuredwith a six-component strain-gage balance located internal to the model.The model is attached to the balance, and the balance is attached to theend of the supporting sting as illustrated in Fig. 1. Both the sting andthe balance will defle ct under aerodynamic and inertial loads.

Deflected Sting

Model~ LSling

~alance~+81/+FN

________ Ir · +PM-----....--/ +zl

-----....f Model

Center-of- ----.............. ........ _______. _S~or ~ ___Rotation ~ ......

Undeflected Sting Position ~

Figure 1. Simple sting deflection geometry.

7

jI

AEDC-TR-76-41

Many models and stings will be subjected to a rapid application ofload during the tunnel starting process and will suffer large static de-flections under certain important test conditions due to the magnitudeof the load applied. Figure 1 illustrates typical sting deflection geome-try. For certain test conditions~ the sting tip angular and translationaldeflection could be several degrees and inches~ respectively. These de-flections are generally in excess of sting deflections normally encounteredin lower dynamic pressure facilities. However~ it is nofthe static de-flection alone which causes concern during the portion of the run whendata are obtained. The oscillatory deflection of the sting~ superimposedon the static deflection which is also present~ can have a major influenceon the accuracy of steady-state data (Ref, 2),

This investigation was undertaken primarily to evaluate the struc-tural dynamics of sting-model systems subjected to transient load con-ditions. The results are applicable to existing test conditions such as:

1, Blowdown or Short-Duration Facilities:The highly impulsive flow conditionscreate sting oscillations which caninfluence data acquisition and accuracyand cause structural integrity loss.

2. Facilities Utilizing Model Injection Systems:The sting-model injection into the tunneltest section under steady flow conditionscan create rather large sting deflections.Here~ the primary concern is for thesting-balance structural integrity ratherthan data accuracy.

3. Facilities without Model Injection Systems:Tunnel starting and stopping loads on themodel and sting are particularly severe fortransonic and supersonic facilities because~of the normal shock which traverses thetest section.

4. Testing with Very High Angles of Attack:High angle-of-attack stall and buffetingof airplane models creates sting transientdeflection conditions and the associatedpr-eviously mentioned problems.

8

AEDC-TR-76-41

5. Facilities Utilized for Dynamic StabilityTesting: During dynamic stability testingfor aerodynamic damping derivatives.the basic sting oscillatory motion can besuperimposed on the dynamic balanceos cillatory motion. This creates dataacquisition and accuracy problems (Refs.3 and 4).

This report discusses various sting materials and describes theanalytical techniques utilized to compute sting deflections. spring con-stants. natural frequencies. and transient motion. Practical utilizationof these computational schemes is demonstrated by comparing experi-mental model sting oscillatory data from a hypersonic impulse tunnelwith computed results.

2.0 ANALYSIS

2.1 STING MATERIALS

Slender beam deflection is primarily caused by bending moments,and the deflection is inversely proportional to El. where E is Young'sModulus and I is the cross-sectional area moment of inertia. Sheardeformation is usually less than 1 percent of the total deflection, unlessthe beam is very short relative to its diameter ( L :::::;d). The allowablebending stress (cry) is also a critical beam design item which, for agiven moment. is directly proportional to dll or 11 d3 .

The wind tunnel test section size limits the model size. The sting-balance cross-section dimensions are dictated by model size limitationsand aerodynamic sting-model interference considerations. Since thesting cross-section diameter is thus essentially specified. additionalstiffness and lower stress levels cannot always be gained by arbitrarilyincreasing the diameter and moment of inertia (I). Consequently. anyreduction in sting deflection must eventually come from inherentmaterial stiffness parameters. which in this case is obviously Young l sModulus of Elasticity (E).

Traditionally. high strength alloy steels have been the primarystructural ingredient for wind tunnel stings and balances. However. inorder to combat excessive deflection, other materials, which are stiffer

9

AEDC-TR-76-41

than steel, warrant serious consideration. Table 1 (Refs. 5, 6, and 7)lists pertinent characteristics of alloy steels and additional materialswhich are potential candidates for sting construction. It is acknowledgedthat the alloy steels are cheaper and easier to fabricate, than any of thestiffer materials.

Table 1 indicates that the sintered tungsten-titanium carbidealloys are about three times stiffer than the steels. However, theyare somewhat marginalso far as the yield stress (cry) is concerned.They also tend to fail in a brittle manner when the yield stress (essen-tially the elastic limit stress) is exceeded. Even so, they merit seriousattention as a primary sting material because of the threefold stiffnessincrease for cases in which sting deflections pose problems.

Table 1. Characteristics of Sting Construction Materials

Material

*Aluminum Alloys*Titanium Alloys

. *N1ckle-Base Superalloys*Steel Alloys

*Cobalt-Base Superalloys*Molybdenum Alloys

*TungstenSintered Tungsten-Titanium

Carbide Alloys

E x 10-6

Ib/in. 2

10.4

16.0

29.0

27.0 to 30.0

33.6

46.0

59.0

**90.0***82.2

w,Ib/in. 3

0.101

0.170

0.296

0.281

0.320

0.370

0.700

**0.525***0.510

-30 y x 10 ,

Ib/in. 2

73

170

175

250

116

125

220

124****

*Ref. 5**Ref. 6, Nominal Properties

*** Ref. 6, Conservative Estimate**** .Estimated from We1bull's formula (Ref. 7)

10

AEDC-TR-76-41

2.2 STING DEFLECTION, STRESS, STRAIN, ANDNATURAL FREQUENCY

Because of model space limitations and aerodynamic sting-modelinterference effects, sting configurations normally consist of one ormore linearly tapered sections combined with one or more untaperedsections. The balance strain-gage lead wires are generally broughtout through a hole down the center of the sting. Thus, cross-sectiondimensions will usually not be uniform throughout the length of the sting.These numerous cross-section discontinuities make closed form inte-gration of the basic slender beam elastic curve equation (Eq. (1)) im-practical for realistic stings. The basic slender beam elastic curveequation is

(1)

where M is the total bending moment at x, and E and I are usually func-tions of x also. Consequently, numerical integration of Eq. (1) is re-quired for practical sting configurations. Because stings are normallycantilevered from some rigid structure, they are ideally suited foranalysis by the first and second area moment propositions (Ref. 8). Thearea moment analysis was incorporated in a computer program whichwas written for the cantilever beam schematically illustrated in Fig. 2.This beam has four different cross -section discontinuities. Each sectioncan either be a constant diameter cylinder or a linearly tapered cone

ElWIRHI

E2W2RH2

E3W3RH3

E4W4RH4

E5W5RH5 R(NEND)

N3

R(NI)

NI N2

---1-"- L3 -r L4N4

-t- L5N5

T2 R(NEND)

-LNEND

Notes: R(Nl)---R(NEND) "Radius at Section DiscontinuitiesNI---NEND " Station Numbers at Section Discontinuities

WI---W5 "WeightNolume of Material in Each SectionEl---E5 "Young's Modulus of Material in Each Section

RHI---RH5 " Radius of Hole through Section

Figure 2. Beam geometry and basic input for STING-1 program.

11

AEDC-TR-76-41

frustum. Each section may have a hole of arbitrary constant diameteralong the centerline. The material of each section is also arbitrary(i. e. steel, aluminum alloys, carbide alloys, etc.), The cross-section shape must be a circle or concentric circles, although this isnot a requirement of the basic beam analysis.

This computer program, STING-l, can perform the followingcalculations after the basic geometric and material properties areprovided.

1. Compute deflections (angular and translational) along the beamlength caused by the weight of the beam. This information is the staticdeflection curve of the beam alone and is necessary for the computationof the first natural bending frequency by Rayleigh's Method (Eq. (4»).Maximum bending stress and strain (along the beam length) caused bybeam weight are also computed by the following standard relations,respectively:

(2)

(3)

(4)

2. Deflection, stresses, and strain caused by concentrated loadsor moments at any station along the beam may be computed. In additionto obvious design applications, the deflections caused by concentratedloads are necessary to compute the beam spring constants or influencecoefficients (see Section 2.3). Figure 3 shows bending stresses fora sting with a concentrated load of 24, 000 Ib at one end computed by CodeSTING-1. Table 2 lists deflections for steel, carbide, and a compositecarbide-steel (C-S) sting computed by STING-l for these sameconditions.

3. The first natural frequency may be computed by Rayleigh's Meth-od (Ref. 9). The effect of concentrated masses (the model for instance)can also be included, The expression for the sting natural frequency is

fL M2 dxo EI

where M is the total bending moment (including beam weight and con-centrated weights) at station x, z is the static deflection at station x, z.is the static deflection at the concentrated weight locations (xi)' and dni

12

AE DC-T R-76-41

is the differential mass of the beam at station x. When the static de-flection curve is used in Eq. (4» the computed frequency will be slightlyhigher than if the true dynamic deflection curve is used. This occursbecause the small deviation of the static deflection from the first modedynamic deflection imposes some constraint which makes the beam morerigid and its frequency higher. This is illustrated in Fig. 4 wherefrequencies for the carbide-steel (C-S) sting are given; both for theRayleigh method with static deflection and the analysis of Section 2. 4where the true dynamic mode deflection shape is computed. A compari-son shows that the Rayleigh method frequency is approximately 6 per-cent greater than the true dynamic mode frequency. Figure 4 also illus-trates the influence of material and model weight on sting naturalfrequency. The natural frequency of mono-material beams (withoutconcentrated weights) scales according to:

Using material properties from Table 1,

(5)

24,000 Ib

wCarbide

W Steel

d = 3.85 in.

All Dimensions Are in Inches

(90)(10 6 )(0.281)

(30)(10 6 )(510)

ID=l.Oin.

1.28

7.48 degd = 10.00 in,

162.0

120

"'" 100Ia~

x'v; 800-

vi 60VI.....-Vl

0> 40c"0C 20

CCI

20 40 60 80 100Station, x, in.

120 140 160

Figure 3. Maximum static bending stress in the sting.

13

AEDC-TR-76-41

Table 2. HIRT Sting Deflections and Natural Frequencies

End Rotation, End Deflection, Frequency, cpsMaterial deg in. (Fig. 4, WM = 0)

Steel 7.37 8.30 17.0

carbide 2.47 2.74 .21.5Composite 2.88 3.81 16.0(Steel andcarbide)

To compute the above information, the following material proper-ties were used:

Material

Steel

carbide

E,lb/in. 2

30(106 )

90(106 )

w,lb/in. 3

0.281

0.525

Steel and Carbide(STI NG-l Results)

2.46 deg

Carbide (STING-l Results)

W~48.5L73.0 --L-32.o 48.5 121. 5 162.0 in.

Note: The composite sting is steel fromstation 121. 5 to station 162. O.

(J

(

~\\\\\'\

\".\~, .~

'--...L ~ ....

I" ... .....::::::::---" --...... -

......- ---------Steel and Carbide ----7;;;(STING-l(MATVEC Results) Results)

4

24

20

II>Q. 16u3"

~c:CI> 12:::lliS....u..

"iii....:::lm 8z

2,000500 1, 000 1, 500Model Weight, WM, I.b

OL-----'-------I..------d------Io

Figure 4. Lowest natural frequency of the example sting forvarious model weights and sting materials.

14

AEDC-TR-76-41

The ratio from the more exact Eq. (4) (WM = O) is:

UJ Carbide

UJ Steel

21.5

17.0= 1.26

4. Certain structural dynamic analyses require the beam to beconsidered as a series of lumped masses connected by springs. Theselection of a particular beam portion or section to form a concentratedmass is a decision requiring some discretion for best results. Oncethe partitioning selection is made, the STING-l program will computethe value of the lumped mass (mi), its mass moment of inertia (IYi) ,and its centroidal position (xi)' This centroidal position is the pointat which concentrated unit loads and moments will normally be appliedto compute the beam spring constants or influence coefficients.

2.3 INFLUENCE COEFFICIENTS (SPRING CONSTANTS)

The structural dynamic analysis described in the next two sectionsrepresents the sting-balance-model system as a series of "lumpedmasses" or concentrated masses. These masses are considered to beconnected together by flat weightless springs. A load applied to onemass will influence the deflection of the other masses because of thespring connections. Consequently, the associated spring constantshave become known as structural influence coefficients.

References 10 (pp. 17-22), 11 (pp. 19-27), and 12 contain excellentdiscussions on structural influence coefficients which basically can beclassified in one of two categories as:

1. Flexibility influence coefficients, defined by:

NL COO Q.

j=1 . IJ J

where

N number of generalized forces applied

qi deflection (translational or rotational)at point i

Qj Forces (or moments) applied at point j

C.. flexibility influence coefficientsIJ

deflection/unit load

15

(6a)

AEDC-TR-76-41

Equation (6a) can be written in matrix form as follows:

(6b)

Basically, Cij, corresponding to the location of the lumped massesof the beam, can be computed from the deflection (translational androtational) caused by unit concentrated loads or moments applied atthe various lumped mass cg locations (xi).

For computational purposes, it is important to note the four differ-ent types of elements which make up the Cij matrix (see Ref. 10, p. 22).The Cij matrix can be partitioned as follows:

where

[ CTFiiJn [ CTMiil

[Coo]I] N

[ CAFiit [ CAM ii] nN = 2n

(7)

CTF ..I]

translational deflection at i caused by unitforce at j, ft/lb

CTM .. = translational deflection at i caused by unitI]

moment at j, ft/(ft-lb) = lIlb

CAF ..I]

angular deflection at i caused by unit forceat j, rad/lb or l/lb

angular deflection at i caused by unit momentat j, rad/(ft-lb) or lI(ft-lb)

This allows Cij to be computed by systematic application of theSTING-1 program to each of the concentrated mass positions.

2. Stiffness Influence Coefficients, given by

NQi = 2: K.. q.i=L I] ]

16

(Sa)

AEDC-TR-76-41

whereq.

J

Q.1

K..IJ

deflection (translational or rotational) atpoint j

force (or moment) applied at point i

stiffness influence coefficients

load/ deflection

In matrix form, Eq. (8a) is written as

K ll K 12 KIN ql

K21 K 22 K2N q2(8b)

---------- --

------------

KN1 KN2 KN qN

Note that Kij is completely analogous to the spring constant, K = lb/ in.employed in one-dimensional spring analysis.

The Kij coefficients are difficult to compute directly from beamtheory. Consequently, they are generally computed by inverting theCij matrix. This is the manner Kij was obtained in the present analysis:

(9)

Another important property of the Cij and Kii matrices is that both aresymmetric. The Cij matrix as computed by :::>TING-1 was not symmet-rical after the third or fourth decimal places (round off error, approxi-mation, etc.). Thus Cij had to be made symmetric so that it wouldproperly invert and yield the following multiplication check:

(10)

where [I] is the identity matrix

It is important to note that the Cij coefficients can be determinedexperimentally (Ref. 12). In fact, thIS is preferable to computing themanalytically and may be absolutely necessary for complex structures.

17

AEDC-TR-76-41

In any case~ some experimental Cij data are valuable for verifying andcomplementing theoretical results. The VKF balance calibration labo-ratories currently measure deflection under loads applied at the modelattachment point so that static sting rotation from model airloads maybe accounted for in data reduction. This type of experimental informa-tion was essential to the application which will be discussed in Section 3.2.

2.4 STING NATURAL FREQUENCIES AND MODE SHAPES

The first natural bending frequency may generally be computed withsufficient engineering accuracy by the Rayleigh method as indicated inSection 2.2. For design purposes~ primarily to avoid resonance condi-tions~ the higher natural frequencies are occasionally required. Refer-ence 9 (pp. 169-170) outlines a procedure by which the influence coeffi-cient (CTFij) in conjunction with the lumped mass concept can be uti-lized to compute the first and higher natural steady-state bending fre-quencies and mode shapes. From Section 2. 3~ the CTFij coefficientsare defined as translational normal deflections at Xi caused by a unitforce applied at Xj. The final matrix formulation is:

CTF 1nffin 2 1 21

CTF 2nffin 22 2 2(11)

=A

CTFnnffin 2n 2n

where .A = 1/ w2 • This is the standard matrix eigenvalue problem where.A is the eigenvalue and the associated eigenvectors (zi) define that par-ticular normalized mode shape. Once the [CTFijmi] matrix is formu-lated~ the eigenvalues (frequencies) and associated eigenvectors (nor-malized mode shapes) may be computed by the MATVEC program (Ref.13), Concentrated weights such as the model may also be taken intoaccount by proper modification of the appropriate mi (see Figs. 4 and 5for MATVEC results for the C-S sting), The mode shapes illus-trated in Fig. 5 are typical for cantilever beams.

2.5 LAGRANGIAN FORMULATION OF STING-MODELMOTION EQUATIONS

Sting-balance-model systems tested in blowdown or short-durationwind tunnels are exposed to transient aerodynamic loads which can act

18

AEDC-TR-76-41

on both model and sting. Since useful data taking time is limited,modern blowdown facilities are designed to vary the sting pitch angle(Cl'p) during the run so that the quantity of data obtained is increasedContinuous-flow tunnels may also have this capability for similar reasons.In addition to sector transient rotation capabilities, some modern faci-lities are capable of translating the sting-model from a location outsidethe test section into the flow field of the test section via a so-called"injection" system.

d • 7.90 in.d' 3.85 in.

WM' 32.2 Ib

o 20 40Carbide --+--- Steel

60 80 100 120 140Station, x, in.

160

1.0~ First Mode! 0 ~~zmax _-e-----e---e- WI. 14.57 cps-1.0 __-e--

1. 0tz Third Mode. ---.......---....

! 0 ~- ~~~zmax ~~ ~::::::::;::::;:~-..,-- W3· 137.62 cps

-1.0 ----------------------=--

Figure 5. Basic bending mode shapes for the examplecarbide-steel sting.

Consequently, any realistic analysis of sting structural dynamicsmust account for elastic deflection of the system caused by:

1. Transient air loads on both model and sting.

2. Transient loads associated with rigid body motionof the system.

To investigate the general transient planar motion of a sting-modelsystem, it has been represented by a series of concentrated massesas illustrated in Fig. 6a. These concentrated masses are connectedby springs so that the motion of one mass influences the motion of the

19

AEDC-TR-76-41

other masses. Because of the rigid body motion of the sting, it isnecessary to employ three basic axis systems which are illustrated inFig. 6:

Inertial axis, XI is parallel to V00'

This axis is fixed to the unde£lected stingand moves with it as shown.

This axis is fixed to the ith concentratedmass and moves with it as shown.

Lagrange's formulation of dynamics (Refs. 14 and 15) is employedto derive the equations of motion for the i th mass. Lagrange's methodprovides a methodical and systematic approach which is especiallyvaluable in the present application involving two moving non-inertialcoordinate systems.

The present formulation is an extension to the example of thecantilever beam and single concentrated mass given in Ref. 11 (pp. 40-41). The present analysis is also similar, in some respects, to theclassical example of airfoil flutter given in Ref. 11 (pp. 210-212) andRef. 10 (pp. 532-536) where Lagrange's formulation is employed.

--_.. Voo

ZI I r-I np.rtiillL::AX;'XI

Detlected

a P I / S'~_I I / I

~ - ~(_~ / Sj / a P--'r--a ,-- ~~ /

\ VOO X~-:.,:z~~~~ r / mi. IYiCenter-ot-Sector Rotation/~ f ~;_--.' ""

//i~-~/~"

Lundetlect:~~"",. "Sting XIII.

I

a. Planar deflection geometry of the sting-model concentrated mass representationFigure 6. Sting coordinate systems.

20

AEDC-TR-76-41

2.5.1 General Lagrange Equation

The Lagrangian formulation of dynamics provides a systematicapproach that gives directly the equations of motion in whatever coor-dinate system may be convenient (Ref, 14). Application of Lagrange'sprocedure is remarkably straightforward even for relatively complexsystems involving moving non-inertial coordinate systems. Lagrange' sbasic equation is

d (aTJ aT au Qkdt aq - aqk + aqk'~.~

'--------~'~

(12)In e~tial fore e s Conserva- Non-conservative

(or moments) tive forces forces (or moments)

(or moments)

where:

k

NN

n

I---N

2n

number of degrees of freedom or independentcoordinates

number of lumped masses in the systemrepresentation

generalized independent coordinate. In the presentanalysis, qk will either be angular rotation or a

linear translation distance, both of which are easilyvisualized physical quantities. The motion direc-tion specified by Eq. (12) is in the exact directionassociated with qk (Ref. 15, pp. 157-158)

T

U

total kinetic energy of system and must be mea-sured relative to an inertial axis system(Ref. 14, p.33)

potential energy of the system. In the presentapplication, U is the W>rk done by the stingacting as an elastic spring.

non-conservative generalized, applied force(or moment) acting .on the i,~,mass,(m) corre-sponding to the k!~ generalized coordinate. Inthis application, Qk is primarily of aerodynamicorigin.

t = time, sec

The appropriate formulations of T J U, and Qk are given in thefollowing sections.

21

AEDC-TR-76-41

2.5.2 The Kinetic Energy and Its Derivatives

In applying Eq. (12) to the sting-model system of concentratedmasses as represented in Fig. 6, a very crucial item is the evaluationof the total kinetic energy and its associated derivatives. These yieldthe inertial forces (or mOJ;nents) of the system. The total kineticenergy for this system is

nT = 2, T.

i=l 1(n = number of masses) (13a)

(l3b)

(14a)

T i is the kinetic energy of the mass (mi) and Vabs is the absolutevelocity of an element of mass (dm), which is located as shown in Figs.6b and c with respect to the inertial axis (XI, 2r). The coordinatesof the particle (dm) with respect to the XI' 2 1 origin are

XI X + xII cos a + zII sin apOi PiP

+ xIII. cos (a p + ei ) + zIII. sin (a p + ei )1 1

InertialAxis

Z - XII sin a + zn cos aa i PiP

---" Voo

ffij

(14b)

b. Coordinates of mass element with respect to moving axis systemsFigure 6. Continued.

22

I nertial Axis

----" Voo

AEDC-TR-76-41

/

LocalDeflected

" Sti ng Axi s"" at Poi nt i

"lip XIII'

I

XI

c. Coordinates of mass element with respect to an inertial axis systemFigure 6. Continued.

Differentiating Eq. (14) with respect to time yields the absolutevelocity components of the particle (dm). Note that xIII, and zIII. =0

1 1since the XlIIi' ZIIIi axis is fixed relative to mass (mi) and rotates withit through the angle (Q:'p + 8i). The origin of the XlIIi' ZIII' axis is notnecessarily the center of gravity of the mass (mi) (Fig. 6cY. Note alsothat xII' is a constant distance which denotes the location of the XIII" ZIII.

1 1 1axis origin along the sting length. Strictly speaking. xII' does vary slight-

_1ly as the sting deflects under load. but this very small change will be ne-glected, The final relation for the total kinetic energy is

T =n 2 2 2,2~ {lloz [.it + Z - 2x x. a' sin a - 2 Z x. 'ap cos a p + x ai:-l a a alp pal i P

+ 2 qi (xasin a p + za cos a p - xi~) + q} + 2 qi (xaap cos a p- za a

psin a

p) + q~a~]mi - [xa sin (a p + qi+n) + za cos (a p + qi+n)

+ (qi - xi ap) cos (qi+n) + qi ap sin (qi+n)](a p + qi+n)(xi m)

+ [xa cos (a p + ql'+n) - Z sin (a + q. ) - (q. - x. a ) sin (q.+ )a p Hn IIp 1 n

(15)

23

AEDC-TR-76-41

where

qi ZII. z.11

qi+n e.1

x. XII.11

(see Fig. 6d)

(see Fig. 6d)

(see Fig. 6d)

(16a)

(16b)

(16c)

(17a)

(see Fig. 6 c) (17b)

(see Fig. 6c)(17c)

(17d)

Note that Xo and 20 are the translational velocities of the model supportsystem moving as a rigid body. This motion will occur. for instance.when a model is injected into a wind tunnel on a system in which themodel. sting. and support translate in unison.

---i.. P ~l Sting--~------rt--.. Vco"', ::--} .

-., --, ~i mi 8

~j

Undeflected znSting

d. Final coordinates of deflected sting with respect to undeflected stingFigure 6. Concluded.

24

AEDC-TR-76-41

Straightforward evaluation of the kinetic energy derivatives yieldsthe following results:

Force Inertial Terms (k = i)

d(aT)dt Jq.

I

aT = (q. _ x.a.. _ q.o. 2 + z cosa + x sinap)miaq. I I pIp a p a1

[cos (qi+n)(ap + qi+n) - sin (qi+n)(o. p +

AEDC-TR-76-41

It is shown in Ref. 11 that

nl K.. q.j=l IJ J

(20)Q.

1

Force (or moment) at point i in thedirection of qi (see Eq. (7a))

In the present formulation,

auaq.

1

auaqi+u

2nl K·· q.j=l IJ J

Sting elastic restoring force at point xi

2nl K. . q.j=l Hn,J J

Sting elastic restoring moment at point xi

(21a)

(21b)

2.5.4 The Non-Conservative Generalized Forces

Equation (12) provides for the arbitary representation of time varyinggeneralized forces (Qk) that may drive and/ or damp the system motion.Qk is a non-conservative force or moment which essentially means thatit is not derived from a potential function such as potential energy (Sec-tion 2. 5. 3). For example, dissipative effects such as sting-model aero-dynamic damping and sting structural damping are clearly in this cate-gory.

2.5.4.1 Unsteady Aerodynamics

In the present application, the primary driving force (Qk) is creat-ed by unsteady aerodynamic phenomena. The resulting aerodynamicloads may be obtained by

1. Theoretical,analysis - Refs. 10 and 11 discuss andcritique the available theoretical efforts. Basictrends are reasonably well defined for ideal config-urations.

26

AEDC-TR-76-41

2. Experimental measurements - Good experimentaldata are highly desirable and should be utilizedwhen available (see Section 3.2).

3. Approximate analysis utilizing both theoretical andexperimental information - Practical requirementsfor unsteady aerodynamic simulation of realisticconfigurations may dictate such an analysis.

The following quasi-steady approximations are used in the presentformulation. For k = i, the aerodynamic normal force is

(22)

where qoo may vary with time and CNTi is composed of both a "static"and "dynamic" contribution as follows:

Stati c ForceCoefficient

CNa,(::J ,eNq, CD~

Dynamic ForceCoefficient

(23)

If nonlinear, then CN' is curve fitted to a suitable function of aT'. Ina similar manner for\: = i + n, the aerodynamic moments are 1

where

then,

StaticMoment

a S. d. C'= I I m T .

I

aerodynamic pitching moment of mass

(mi) at point (Xi)

(aT.d)+ Crn .. 2f- +a l 00

Dynamic Damping Moment

27

(25)

(26)

(27)

AEDC-TR-76-41

Also~ if Cm.(aT.) is linear~ then1 1

(28)

Otherwise~ it is curve fitted as a function of aT" Now aT'~ the instan-1 1

taneous angle of attack at station xi along the sting~ is:

~AngularRotation '.

Xi a p -

Xi ---l

AEDC-TR-76-41

Inclusion of the angular rate (aT' and qr') dependent aerodynamicforces and moments has a negligible leffect oil. motion amplitude anddecay for the relatively stiff stings commonly employed in standardstatic stability and control testing. However. attention must be focusedon their influence in dynamic stability testing when a highly sensitivedynamic balance is utilized.

~lrDeflected Sti ng ZIJZ'I f

zI xiSI

Sting

a P Ta nge nt to DeflectedSting at Xl = 0

Axial Force Moment =AFI [XiSI - (zI - li) ]

Figure 7. Axial force bending moment schematic.

2.5.4.2 Structural Damping

Observation of experimental sting motion data obtained from theVKF Tunnel F (Section 3.2) revealed that the motion was highly damped.This timewise decrease in amplitude was much more than could beaccounted for by aerodynamic damping alone. An investigation revealedthat the decrease in amplitude was essentially linear with increasingtime. This manner of amplitude decay is normally associated withCoulomb or friction damping (Ref. 9) and the single degree of freedomdifferential equation is of the form:

fiX + Kx = -F __X__I xI

29

(33)

AEDC-TR-76-41

F is a constant friction force and always acts to oppose the motion.Consequently, it was suspected that some form of structural dampingcapable of dissipating a large amount of energy was prevalent in theVKF Tunnel F sting.

A brief literature survey revealed that there are basically twotypes of structural damping:

1. Internal damping of a solid homogeneous structuralmember caused by internal grain structural, plasticstrain, magnetic-electrical, or thermal effects(see Ref. 18 for a comprehensive survey), and

2. Damping of joints, connections, or interfaces ofthe members of a total structure caused by inter-facial Coulomb friction and slip (see Refs. 19 and20).

Reference 20 reports that under optimum circumstances, theenergy dissipated in joint slip damping may be large, not only comparedwith internal damping, but also compared with the total strain energyof the structure. Since the VKF tunnel-sting contained several promi-nent joints and its motion was highly damped with essentially a lineardecay, it is concluded that damping is primarily caused by interfacialCoulomb friction.

Consequently, it is postulated that the sting damping could bemodeled in a rather gross manner by the following relations which aresomewhat analogous to the Coulomb friction force in Eg. (33). Thedamping force at Xi is

2n • qiQDF i = -FCOF i j~l K ij qj TD

The damping moment at x. is:1

QDM j = -FCOM j I~~l Kj+n,j qjlwhere

2nI K.. q.j=l 1]]

and

2nI K· . q.

j=1 I+n,J ]

30

(34a)

(34b)

AE DC-TR-76_-41

are the sting elastic restoring force and moment, respectively, at xi(Section 2.5.3). FCOFi and FCOMi are arbitrary constants which areutilized to adjust the fractional magnitudes of the sting elastic restoringforces and moments, respectively, which are resisting the sting motion.They are determined by numerical experimentation and comparisonwith experimental sting motion damping data (see Section 3.2).

Formulation of a more elaborate mathematical model of stingstructural damping than Eq. (34) will require both analytical and experi-mental results for typical wind tunnel sting joints.

2.5.5 Specific Motion Equations

The specific motion equations of each individual concentrated mass(mi) of the sting-model system for each of its two degrees of freedomcorresponding to the general formulation (Eq. (12)) can now be writtenby combining appropriate results from Sections 2.5.2, 2.5.3, and2.5.4.

For k =i, Eqs. (18a), (21a). (22), and (34a) are substituted in Eq.(12) to yield the force relation (in the translational zi direction) for mass(mi). It is convenient for future purposes to write this equation asfollows:

where:

(RHSF)i (35a)

(RHSF\

m·I

- [cos (q.+ )(x. m.) + sin (q. )(2. m.)]I n I I l+n I I

(x. a + q. 0. 2 - Z· cos a - x sin a ) m·Ip Ip 0 po pI+ [cos (q. )(2. m.) - sin (q. )(x. m.)](q· .l+n I I l+n I I l+n

2n- 2, K.. q. + NT. + QDF ij=1 IJ J I

(36a)

(36b)

(36c)

For k = i + n, proper combination of Eqs. (18b). (21b), (25) and(34b) yields the moment equation (in the rotational ei direction) for mass(mi) . This is:

31

AEDC-TR-76-41

(RHSM)i (35b)

where

(36d)

(36e)

(RHSM)I' = [(z - x. a - q. a2) cos (q. ) + (x + 2q·. ao I pIp l+n 0 I P+ qi Up - Xi a;) sin (qi+n)] (xi mi) + [(zo - xi ap - qi a;) sin (qi+n)]

.. .2 _ 2n- (xo + 2qi ~ + qi a p - xi a p) cos (qi+n)](zi mi) - j~l Ki+n,j qj

- AF1[x/ql+n) - (ql '-- qi)] + MT . + QDM iI

(36f)

Equation (35) is comprised of the basic equations which are bothstatically and dynamically coupled together in the general case. NTiand MT' are still in arbitrary form. If NT. and MT., as representedby EqS.l(22}, (23), (26), (27), (29), (3D), ahd (31), eire substitutedappropriately, then:

(37a)

- [cos (q. )(x. m.) + sin (q. )(z. m.)]l+n I I l+n I I (37b)

(RHSF\ (.. . 2.. .. . )

Xl' a p + q. a - Z cos a - X sm a m·lp 0 po pI

(x. a - q.)Y. I P I 00

y 2 co sa00 p

Z Vjo 00y 2

00

2 2n+ [cos (qi+n)(Zi m) - sin (qi+n)(xi mi)](qi+n + ap) - j~l Kij qj

(xi ap - qi)(a p sin a p)+---''------:::-''---_....:.....

Y cos 2 a00 p

32

Y cos a00 p

- cos (q. )(x. m.) - sin (q. )(2'. m.)l+n 1 1 l+n 1 1

AEDC-TR-76-41·

(37d)

(37e)

[(2 0 - xi a p qi a;) cos (qi+n) + (xo + 2qi ap + qi ap- x. a2) sin (ql'+n)] (x. m.) + [(2 - x. a - q. a2) sin (q. + n)Ip 11 DIp Ip 1

z V]o 00+--y 2

00

+ (Ma. + Mq,l1 1

(37f)

Note that FNi (aT) and PMi(aT) are the "static" aerodynamic force andmoments, respectively, which may be nonlinear functions of aT andother variables as well. In general, curve fitting or table look-upinterpolation of these quantities will be required.

2.5.6 Numerical Solution

Equation (35) is a system of 2n second-order differential equationswhich must be solved simultaneously. The general complexity is suchthat a numerical solution is required. To accomplish this, a computerprogram (RKAM) documented in Ref. 21 was employed. In order toapply this computer program written for simultaneous first-order dif-ferential equations, it is first necessary to solve Eq. (35) for qi andqi+n explicitly. This yields

m2 2 .(RHSF)i - m1 2 .(RHSM)i1 1

Z.1

33

(38a)

AE DC-T~ -76-41

ffill.(RHSM)j - ffi21.(RHSF)j1 1

e.1

These are 2n second-order equations of the general form:

qk = fk(qj' Crj" t) {i = 1 --- 2n}

k = 1 --- 2n

where the initial conditions are:

(38b)

(39a)

{

k = 1 --- 2n}

k = 1 --- 2n(39b)

Equation (39) can be reduced to a system of 4n first-order equationsby the following substitution:

Then

k 1 --- 2n (40a)

k = 1 --- 2n (40b)

Equation (40) is a set of 4n first-order equations to be solved simul-taneously subject to the following initial conditions from Eq. (37b):

k

k

1 --- 2n

1 --- 2n(40c)

Equations (40a). (40b). and (40c) are in the form required forsolution by the RKAM program (Ref. 21). The resulting computer pro-gram is hereafter referred to as SMD2NDOF.

The sting rigid body motion variables (Qlp. lip. Gp • *0' xo' zo. andzo) must be known as a function of time. These variables are requiredif the sting system moves as a rigid body (for example. during.modelto tunnel injection or sector rotation). Va:l and. occasionally. V 00 andqlD must be known as a function of time. These transient variables arecurrently curve fitted by a Fourier Series or a least-squares polynomialand can thus be computed in SMD2NDOF when needed as a function oftime. A table look-up interpolation representation may also be employed.

34

AEDC-TR-76-41

3.0 APPLICATIONS

3.1 HIRT STING DESIGN AND TRANSIENT RESPONSE

References 22 and 23 provide information concerning the AdvancedTransonic Transport or the Advanced Technology Transport (ATT)model which is a representative candidate for analysis. Figure 8 illus-trates the predicted history of the dynamic pressure in a pilot l highReynolds number l Ludwieg-type wind tunnel. This information isbased primarily on experience with such a VKF pilot facility (Ref. 1).Maximum steady-state dynamic pressures (Clocm. ) were selected fromconditions reported in Ref. 22. ax

-

1.00.80.4 0.6Time, t, sec

0.2

~St~rti ng-----lr- Time ·1

oo

0.8

0.6

1.0

0.4

O. 2

Figure 8. Predicted variation of dynamic pressure ratio in a PilotLUdweig-Tube Wind Tunnel.

Both the steel sting and the C-S sting were investigated at criticalsting design load conditions. One critical point was test condition No. 7of Ref. 22 1 Table 2. For this condition l the ATT model at aT of 8. a deggenerates an aerodynamic normal force of 24 1 000 lb (see Fig. 3). This

35

AEDC-TR-76-41

means that Q:'p (the initial fixed preset angle of the sting~ Fig. 6a) plus81 (the angular deflection at the model location) must essentially equal8 deg. Thus~ in order to initially set Q:'p before the run~ 81 (t) must becomputed.

The steel sting exhibited highly unsatisfactory deflection divergencecharacteristics under dynamic load conditions. Just a small initialpreset angle of attack (Q:'p) was enough to cause very large sting deflec-tions before the sting strain energy restoring force became sufficientto prevent further deflection (see Fig. 9). The situation is analogousto the torsional divergence of a wing (Ref. 11~ Chapter 3). A smallincrease in Q:'p from 0.60 to about 0.85 would surely result in catastro-pic sting failure. The situation is so unstable that this ATT modelcould not be tested at this condition with a steel sting.

1.61.4

Remarks--CNa Only

78. 10 ± 0.03

77.21 ± 0.03

1.2

DOF

2

ModelWeight, Ib

305

0.6 0.8 1.0Ti me, t, sec

rEnd of Tunnel Starti ng Time

10.0 Normal Force24,690lb

--------8.0

c 6.0

...:.N

4.0

2.0

0

10.0

Nor mal Force8.0 24,690lb

- --------

6. 0C1Q)"0

...:.cD

4.0

2.0

00

a. a p =0.60 degFigure 9. Motion history for ATT model and steel sting, condition

No.7, basic qoo variation.

36

AE DC·TR ·76·41

10. a Normal Force24, 000 Ib

8. a

a p : 0.50 deg

c 6. a

v,N

4.0 la, 000 Iba p : 0.25 deg

2. a

a

10. a

Normal Force ap = 0.75 deg8.0 24, 000 Ib

6.0 a p = 0.50 degClQ)"0

CIJv,4.0 10,000 Ib

a p = 0.25 deg

2.0

aa o. 2 0.4 0.6 0.8 l.0 1.2

Time, t, sec

b. a p =0.25, 0.50, and 0.75 degFigure 9. Concluded.

Figure 10 illustrates the motion history (81 and z 1 versus time) forthe ATT model and the composite C-S sting. Here, Qip is 5.0 deg, andthe ATT test condition No. 7 is being simulated. Deflection is less thanhalf of that observed for the steel sting (Fig. 9a); however, this is acritical condition for the carbide portion of the sting (x < 48. 5 in. )because of bending stresses at these loads. Results for-two modelweights and two different lumped-mass representations (n = 1 and n = 15)are shown in Fig. 10. For n = 1, note that essentially the same resultsare obtained for both model weights (except for the frequency). The fre-quencies agree very well with predictions for the C-S sting in Fig. 4,particularly the MATVEC results. For n = 15, the sting is representedby 14 lumped masses so that sting inertia is taken into account. This

37

AEDC-TR-76-41

more complex simulation of the sting makes about 15-percent differencein the results which may be enough to warrant inclusion in some detailedanalysis. However, the n = 15 results shown required about 7 hr ofcomputation time on the IBM 370/165, whereas the n = 1 results requiredonly a few minutes.

Figure 11 illustrates what would happen if the dynamic pressurewere a step input for ATT test condition No.7. Results for two modelweights are shown (:s, was held constant). Considerable oscillation ispresent with the heavier model exhibiting the larger magnitude. It isspeculated that the C-S sting would fail catastrophically for this particu-lar step load input condition.

Several computer runs (not illustrated herein) were made with thebasic wind tunnel qco variation (Fig. 8) perturbed so as to yield slightlyfaster initial rise times. The resulting sting oscillation amplitudeswere then somewhat more than for the basic qco variation (Fig. 10) butconsiderably less than for the step qco input (Fig. 11). Both the heavyand light models exhibited practically identical oscillation amplitudes(very low) for the basic qco variation (Fig. 10, n = 1). The basic qco(Fig. 8) aerodynamic loading is then essentially like a "static" loadingsince inertial loads were almost nil. For the step qco input (Fig. 11),the loading was indeed "dynamic" and the heavy and light models didnot oscillate at the same magnitudes because of the strong inertialeffects. It is worth noting that, if the tunnel dynamic pressure risetime is slower than the basic variation of Fig. 8, then the steady-state run time is decreased accordingly, and very little decrease insting oscillation amplitude is gained. Consequently, it is concludedthat the example wind tunnel's basic qco transient variation wasessentially optimum from the viewpoint of both sting oscillation andsteady-state run time for model-sting systems of this size.

In lieu of actual experimental transient data, the aerodynamic loadsfor the ATT were generated from Eqs. (22) and (26) where qoo = qoo (t).Computer runs were made with reasonable estimates of CN ' CN.' CN '. Q' Q' qCm ' Cm ., and Cm . It was found, for this particular model and stingQ' Q' qattachment point, that only the normal force (CN ) influenced the sting

Q'deflection to any practical extent. This behavior was also noted in asting-model motion simulation reported in Ref. 2. These stings are sostiff that aerodynamic damping exerts practically no influence. Grosslyexaggerated static moment inputs (both stable and unstable via Eq. (28))did produce noticeable changes in deflection (approximately += 15 percent,respectively).

38

AEDC-TR-76-41

1.21.0

~ Remarks

5 CNa Only6 CN a Only

10 CNa Only

n DOF

15 301 21 2

r End of Tunnel Starting Time

ModelWeight, Ib

1,2701,270

350

-- 4. 236 ± O. 088

('f'U'1)...C>-{T'(:>"O-o-a""D - 3. 672 ± O. 046

o 3. 666 ± O. 001

0.4 0.6 0.8Time, t, sec

o

0.2

Nor rna I Force25,500 Ib

-23~750lb- ------..,-,,-....---- --3.225 ± 0.065

.cru-o-,.-,..u'"'u-o..o-u"'O - 2. 798 ± O. 035

o 2. 795 ± O. 001

4.0

3.0

c::..:. 2.0

N

1.0

0

4.0

3.0

enQ)"0 2.0-'"

1.0

00

Figure 10. Motion history for ATT model and carbide-steel sting,condition No.7, basic q"" variation, ap = 5.0 deg.

ModelWeight, Ib

3501,270

~ Remarks 11.10.0 CNa Only6.0 CNa Only

DOF

22

0.6

\

0.5

I\qm =qmmaxqm

Time

,....,/ \

I \

0.1

8.0

6.0

c::4.0

N

2.0

0

6.0

en4.0

Q)"0

ci'2.0

0.2 0.3 0.4Time. t, sec

Figure 11. Motion history for ATT model and carbide-steel sting,condition No.7, q"" = qoom a)(' ap = 5.0 deg.

39

AEDC-TR-76-41

3.2 VKF TUNNEL F STING DYNAMICS

Recent results obtained in VKF Tunnel F have provided an opportunityfor comparison with experimental motion data. During this test program,the sting angular deflection'was found to be greater than could be account-ed for by model aerodynamic loads alone. High-speed motion pictures(framing rate ~4000 frames/ sec) were taken during several shots toprovide a record of the sting-model deflection. It was speculated thatthere were significant aerodynamic and inertial loads on the sting itselfwhich contributed to the observed deflection. Consequently, the SMD2NDOFcomputer program was applied to simulate the sting motion, both withand without sting aerodynamic loads. A general description of the VKFHypervelocity Tunnel F is contained in Ref. 24.

The Tunnel F sting-balance was modeled structurally via theSTING-1 program. Some experimental calibration rotational deflectiondata for the balance alone and the balance plus sting were required toobtain a reasonable stiffness representation of this model supportsystem. The basic sting utilized for this particular test is not as stiffas its exterior dimensions would suggest because of numerous jointsand cross-sectional area discontinuities. Figure 12a illustrate's theactual sting used during the tests, including the force balance. Thegeneral complexity of this actual sting force balance system is such thata number of dimensional iterations were necessary via the STING-l pro-gram to obtain the structural stiffness synthesis illustrated in Fig. 12b.A somewhat more complete synthesis of the sting-force balance systemis feasible and should be incorporated in future analyses.

Transient model loads (corrected for inertia effects) were measuredduring the test. These were curve fitted with a Fourier series andinput as forcing functions (NT1(t) and MT1 (t» to the SMD2NDOF program(Fig. 13). Strictly speaking, the forward and aft normal loads measuredby the force transducers in the balance were converted into a normalforce and pitching moment which were assumed to act at a single model-balance attachment point (the physical cg location of the model, Fig. 12b).This pitching moment makes a sizable contribution to the sting deflectionand cannot be neglected.

Figure 14 illustrates the Tunnel F transient flow velocity historyfor this particular test.

40

Model cg

Force Baja nceOD = 2.75 in.ID= 1. 00 in.

OD =4.50 in.ID = 2.00 in.

Sector Hub

+---1+1[--+1-- ~--

OD=6.00in.

All Dimensions in Inches

--~

~......

Model cg

a. Actual sting configurationSector

----~-~~

OD = 3.50 in.

1D=0.50 in.

b. Structural stiffness synthesisFigure 12. VKF Hype~elocity Tunnel F stings.

»mon.:..J:J~C)

.;,.

AEDC-TR-76-41

0.20O. 16O. 12

Experimental

o Fourier SeriesRepresentation(Selected Poi ntsl

Note: MTI Is the Pitching Moment Acting atthe Physical cg of the Model

Time, sec0.080.04

Arc Fired600

500

400.0

,.:.. 300f-;z

200

100

0

60

50

.0 40I-- 30,.:..f-:2: 20

10

00

Figure 13. Transient aerodynamic loads on the Tunnel F model.

Experimental

o Fourier Series Representation(Selected Poi ntsl

5,000

4,000

u 3,000Q)

~:t::

82, 000>

1, 000

00

Arc Fired

0.04 0.08Time, sec

0.12 O. 16 O. 20

Figure 14. Transient flow velocity history for VKF Tunnel F.

42

AEDC-TR-76-41

Since airloads on the sting were to be included~ the sting was sub-divided into 14 sections (concentrated masses~ n = 15~ including themodel). No experimental data were available for the aerodynamic loadsacting directly on the sting itself. Consequently ~ these airloads wereestimated in the following manner. It was assumed that the airloads(NTi) on the individual sting masses (mi) had the same timewise varia-tion as the airload (NT1 ) on the modeL However~ the magnitude of thesting airloads is not equal to model airloads because of different plan-form areas and different shape effects. The sting is composed of aseries of circular cylinders and cone-frustums~ while the model isessentially conicaL The airload normal force on a cylindrical concen-trated mass is then

(41a)

where CN1 (model) and CNi (sting mi) are based on planform area. CNifor the sting was estimated from the Newtonian normal force coefficientfor circular cylinders (Ref. 25~ Chapter 17). The ratio (CN/CN1)remains practically constant from 15 to 17 deg which essentially bracketsthe observed angles of attack. Api and Ap . are the planform areas ofthe model and mi~ respectively. Planform

1areas (Api) corresponding to

the actual sting (Fig. 12a) were employed. For cone-frustum sectionsof the sting~ shape effects were assumed to be similar to those of themodel so that

N (t) = (APi)N (t)T. A T 1

1 PI

Equations (41a) and (41b) have the general form:

(41b)

(41c)

The- loads on the sting section were assumed to act at the lumped masscg and produce no pitching moment (MT.(t)).

1

Only the last 12 sting masses were considered to be aerodynami-cally loaded since the two forward portions of the sting are inside themodel. The summation of Ci (i = 4 to 15) yields the ratio of the stingtotal normal airload to the model normal airload. In this particularcase. 15

I C. = 3.31i=4 1

43

(42)

AEDC-TR-76-41

This indicates that the sting airloads are about three times the modelairloads, Fortunately, the sting loads are distributed primarily alongthe stiffer aft sections which alleviates their deflection contributionconsiderably, The theoretical magnitude of the sting airload contri-bution to the static deflection can be computed via the deflection influ-ence coefficient approach (Section 2,3), Theoretical results fromSMD2NDOF indicate that

e1 {with sting loads}e1 {without sting loads}

1.12

Figures 15 and 16 illustrate both experimental results andSMD2NDOF predictions for this TlU1nel F test. Motion predictionswere made with and without sting aerodynamic loads. Each predictionrequires about 45 min actual computation time on the IBM 370/165.The experimental data (from high-speed motion pictures) are e1 (Fig.15) and znose (Fig~ 16) versus time. znose was computed by SMD2NDOFas illustrated in Fig. 17.

The predicted dynamic oscillation amplitude (oscillation about themean or static level, Fig. 15a) is somewhat greater than the experi-mental magnitude, particularly near the end of the run. It is evidentfrom the experimental data (Figs. 15a and 16a) that considerable damp-ing of the oscillation is occurring. From previous results (Section 3. 1),it is known that the damping is structural rather than aerodynamic.Sting structural damping was included in the SMD2NDOF predictionsillustrated in Figs. 15b, d and 16b, d. The numerous joints of the actualsting (Fig. 12a) make an important contribution to structural damping(Section 2.5.4. 2). This is fortlU1ate from a "static" data accuracyviewpoint and more than compensates for the apparent decrease instiffness introduced by the joints.

The average magnitude (mean value. of the amplitude) of e1 versustime is shown in Fig. 15c. Examination of these mean amplitudesreveal a maximum difference of about 0.08 deg between experimentand prediction with sting loads, and about 0,20 deg difference betweenexperiment and prediction without sting loads. The predicted resultswith sting loads included are in remarkably good agreement with ob-served magnitudes near the beginning of the run. These results showthat it is feasible to account for sting aerodynamic loads on e1 (andhence aT1)'

There is a slight discrepancy between the predicted (48 cps) andexperimental (55 cps) oscillation frequencies. When modeling thesting balance system, the structural stiffness synthesis was of primary

44

AEDC-TR-76-41

concern. and when those criteria were satisfied. wi was 48 cps forsting-balance plus model (STING-l program). This was deemed satis-factory since the system frequency was known to be on the order of 50cps. A large amount of structural damping distorts the sinusoidalamplitude variation considerably (Figs. 15b and 1Gb) and some discre-tion is required when computing the frequency from a limited numberof cycles of experimental data.

Ti me, sec

a. Transient amplitude without structural damping

0.200.16

SMD2NDOF (n = 15) with Sting Airloads,No Structural Damping

---- SMD2NDOF (n = 15) without Sting Airloads,No Structural Damping

o Experi me ntal, High-Speed Motion-PictureData, (J 1~ 55 cps

.Arc Fired2.00

1. 50

I"I \~

0">Q)"C 1.00

..:..CD

0.50

2.00 /\ rc Fi red

O. 20O. 16

o

O. 12

SMD2NDOF with Sti ng Ai rloadsand Structural Dampi ng,FCOFi =FCOMi =0.050

o Experimental High-SpeedMotion-Picture Data

0.08

o

O. 04OL...-...ol-----L_.....---'-_.l---J----l_-'---'-_-'---'----l_-'---J.._-'---'-_'---l-----L---.Jo

1. 50

=Q)"C

1. 00...:-.CD

0.50

Time, sec

b. Transient amplitude with structural dampingFigure 15. Theoretical and experimental angular deflections of the Tunnel F model.

45

AEDC-TR-76-41

cr> CI.) "0

> ro .-<

CD

-CI.)

"0 :::J

:=:

2.00

1. 50

1. 00

. ,-- Arc Fired 0.50

'\ "-" '-,

' ........ . -

Sym

-- SMD2NDOF Mean Value with Sting Aerodynamic Loads (n = 15)

--- SMD2NDOF Mean Value without Sti ng Aerodyna mic Loads (n = 15)

-0- Average Value, High- Speed Motion-Picture Data

---------oL-~~-J--~J-~~--~~~--~~~~--~~~~--~~

o O. 04 O. 08 O. 12 O. 16 O. 20

0.5

Ti me, sec

c. Average amplitude

Sym

o Experimental High-Speed Motion-Picture Data

o SMD2NDOF with Sting Aerodynamic Loads and Structural Damping, FCOFi = FCOMi = O. 050

0.4 Arc Fired

O. 3 Note: Esti mated

Time

9 = 91 - 9 la mplitude max lav

~ 0.2

Experi mental Data T Accuracy is ±O. 15 deg - O. 15 deg ro

cD

O. 1 ~ T o ~~~ __ ~~~ __ ~~~L-~~ __ ~~~~~~~ __ ~~~~ o O. 02 O. 04 O. 06 O. 08 O. 10 O. 12 O. 14

Time. sec

d. Oscillation amplitude FigUre 15. Concluded.

46

O. 16 O. 18 O. 20

IIJII

/ 1. 00

O. 80

~~ 0.60 [

c . N 0.40

I O. 20

0

-0.20 0

Arc Fired

r I \ I \ I I I V

0.04 0.08

AEDC-TR-76-41

Sym

SMD2NDOF (n = 15) with Sting Airloads, No Structural Damping

---- SMD'2NDOF (n = 15) without Sting Airloads, No Structural Damping

o Experimental, High-Speed Motion-Picture Data, (J;;:; 55 cps .

(J 1 :::; 48 cps

O. 12 O. 16 0.20 Time, sec

a. Transient amplitude without structural damping

1. 00 r- .r-Arc Fired

0.80

c: 0.60 ci> ::g

r-f 0.40

0.20

Sym

SMD2NDOF with Sting Airloads and Structural Damping, FCOFi = FCOMi = O. 050

o Experimental High-Speed Motion-Picture Data

o I ..t o O. 04 O. 08 O. 12 O. 16

Time, sec

b. Transient amplitude with structural damping Figure 16. Theoretical and experimental translational model nose tip deflections.

47

AEDC-TR-76-41

--- SMD2NDOF Mean Value with Sting Aerodynamic Loads (n = 15)

----- SMD2NDOF Mean Value without Sting Aerodynamic Loads (n = 15)

--0- Average Value, High-Speed Motion-Picture Data

Time, sec

c. Average amplitude

0.200.160.120.08

'''''' .....,......,

............

---------------O. 04

Arc Fired

1.00

O. 80

c 0.60:>ra

Q)V>0

0.40c:N

0.20

00

Sym

o Experi mental High-SpeedMotion-Picture Data

o SMD2NDOF with Sti ng AerodynamicLoads and Structural Damping,FCOFj = FCOMj = O. 050

Arc Fired

O. 20

Time

0.16

0.015 in..--

Q)V>oc:

N

Znosea mplitude = znosemax - znoseav

O. 12O. 08O. 04

0.015 in.

TNote: Esti mated

Exper ime ntaI DataAccuracy Is±0.015 in.

O. 10

0.08

C.-0., O. 06"t:l:::J

:=:0-Era 0.04Q)

Vl0c:

N

0.02

00

Time, sec

d. OsciUation amplitudeFigure 16. Concluded.

48

AEDC-TR-76-41

Znose

k~~lNC9~- / /ModelCg

I T---~_ Deflected Sti ngCenterli ne at

(~_ '[ ----,~-_-~---rodelcg Location

Undeflected Sting Centerline~ ~Figure 17. Model nose tip translational deflection geometry.

A realistic assesment of the Q'plunge magnitude in Tunnel F is ofmore than academic interest since it is difficult to measure experimen-tally and it does contribute to the total transient angle of attack. of themodel (Q'T 1). From Eg. (29)1

(43)

where in the present case l

v cos a00 p

(44)

Theoretical results (including sting aerodynamic loads and struc-tural damping) for both Q'plun e and zl are illustrated in Fig. 18. Themaximum value of the predicted Q'plunge is approximately one tenth ofa degree. Because of the large amount of structural damping present lit is concluded that the maximum Q'plunge will normally be less than0.025 deg for force tests in Tunnel F when steady-state data are takenafter the initial starting transients.

49

AEDC·TR·76·41

SMD2NDOF with Sting Airloads, No Structural Damping

SMD2NDOF with Sti ng Airloads and Structural Dampi ng,FCOFi = FCOMi = 0.05

O. 80Arc Fired

0.60

c 0.40.-,..:,

N

0.20

0

- O. 20 L..-....I---I-_'----L.---'----'_-'----'---'-_-'---'----I-_'----L.---'---'_-'----'---'-----'

0.20en

AEDC-TR-76-41

To assist in structurally analyzing realistic sting configurations,a computer program (STING-i) was written (Section 2.2). With geome-trical and material properties as input data, STING-l computes deflec-tions, first natural frequency, and influence coefficients (spring con-stants) for cantilever beams with mixed tapered and untapered sections.The material of each section is arbitrary so that a certain type of com-posite sting may be analyzed. Sting design applications are obvious;however, perhaps the most important application so far has been thecomputation of structural influence coefficients (Section 2.3) requiredfor certain dynamic analysis procedures (Section 2.4 and 2,5.3). TheSTING-l program is based primarily on elementary strength of materialconcepts but is general and flexible enough to model most of the morecommon sting-balance configurations.

The MATVEC computer program (Ref. 13) which solves the standardmatrix eigenvalue problem is recommended for the computation of stingsteady-state vibration natural frequencies and mode shapes. The inputrequired is the [CTFij mi] matrix (Section 2,4), where both CTFijand m. may be computed by the STING-l program, Prior experiencein a slructural laboratory with cantilever beams has verified MATVECpredictions,

4.2 STING STRUCTURAL DYNAMICS

Lagrangian mechanics has been employed to derive the coupled dif-ferential equations of planar motion for a rather general sting-balancesystem (Section 2. 5). This system is represented by concentratedmasses which are tied together by spring constants (influence coefficients)and may be subjected to arbitrary known transient forces and moments.The formulation allows sting loads as well as model loads to be included.Also the system is allowed to move (rotate and translate) as a rigid body.Thus with the proper input, transient conditions occurring during modelto tunnel injection and/ or sector rotation may be investigated.

The general formulation is such that a numerical solution is requiredfor all but the simplest, almost trivial, cases. Consequently, a stand-ard program (Ref. 21) for solving simultaneous differential equationswas employed and modified as required. The resulting program(SMD2NDOF) is versatile enough to simulate a wide variety of stingstructural dynamic situations. The sting-model synthesis does not haveto be an elaborate multi-mass system to provide useful information. For

51

AEDC-TR-76-41

example l see Section 3. 1 where a single mass representation was em-ployed for the analyzed sting-model. The simulation of the Tunnel Fsystem (Section 3.2) required a more elaborate multi-mass representa-tion since sting aerodynamic loads were to be included.

REFERENCES

1. Starr, R. F. and Schueler, C. J. "Experimental Studies ofa Ludweig Tube High Reynolds Number Tunnel." AIAAPaper 73 -212, Presented at the AIAA 11th AerospaceSciences Meeting, Washington, D. C., January 10-12,1973.

2. Picklesimer, J. R' I Lowe, W. H., and Cumming 1 D. P."A Study of Expected Data Precision in the ProposedAEDC HIRT Facility." AEDC-TR-75-61 (ADA013814),August 1975.

3. Burt, Glen E. and Uselton l James C. "Effed of Sting Oscil-lation on the Measurement of Dynamic Stability Deriva-tives in Pitch and Yaw." AlAA Paper No. '"i4-612.Presented at the AIAA 8th Aerodynamics Testing Con-ference l Bethesda, Maryland, July 8-10 1 1974.

4. Canu, Michael. "Wind Tunnel Measurement of AerodynamicPitch Damping of a Model Oscillating in Two Degrees ofFreedom." ERD-TT-72-18 (May). Technical Transla-tion of a report prepared for ONERA1 Modane-Avrieux,France.

5. Materials in Design Engineering. Materials Selector Issue.Vol. 64 1 No.9, Mid-October 1966, Reinhold PublishingCompany, New York.

6. Smithells, C. J. Metals Reference Book. Third Edition,Butterworths, Washington, 1962.

7. Powder Metallurgy. Edited by Werner Leszyneski,Interscience Publishers, New York, 1961.

8. Laurson, P. G. and Cox, W. J. Mechanics of Materials.Third Edition, John Wiley and Sons, Inc., New York,1954.

9. Thomson, W. T. Mechanical Vibrations. Prentice-Hall,New York, 1954.

52

AEDC-TR-76-41

10. Bisplinghoff, R. L., Ashley, Holt, and Halfman, R. L.Aeroelasticity. Addison Wesley Publishing Company,Inc., 1957.

11. Fung, Y. C. An Introduction to the Theory of Aeroelasticity.John Wiley and Sons, Inc., New York, 1955.

12. Martin, D. J. and Lauten, T. "Measurement of StructuralInfluence Coefficients." Chapter 1, Part IV of AGARDManual on Aeroelasticity. Ed. by R. Mazet, October1961.

13. Ehrlich, L. "Eigenvalues and Eigenvectors of Complex Non-Hermitian Matrices Using the Direct and Inverse PowerMethods and Matrix Deflation. I' The University ofTexas Computation Center, Publication No. UTF4-01-001, August 2, 1961 (Program MATVEC).

14. Wells, Dare E. "Theory and Problems of LagrangianDynamics." Schaum's Outline Series, Schaum's Pub-lishing Company, New York, 1967.

15. Goldstein, Herbert. Classical Mechanics. Addison-WesleyPublishing Company, Reading, Massachusetts, 1959.

16. Brown, R. C., Brulle, R. V., Combs, A. E., Vorwald,R. F., and Griffin, G. D. "Six-Degree-of-FreedomFlight Path Study Generalized Computer Program. f'FDL-TDR-64-1, Part 1, Volume I, and Part 2, Volume1. Prepared by McDonnell Aircraft Corporation forWright-Patterson AFB, AFFDL, Ohio, October 1964.

17. Brunk, James E. "Users Manual: Extended CapabilityMagnus Rotor and Ballistic Body 6-DOF TrajectoryProgram." AFATL-TR -70 -40, May 1970, Preparedby Alpha Research, Inc., For AFATL, Eglin, AFB,Florida.

18. Lazan, Benjamin J. Damping of Materials and Members inStructural Mechanics. Pergamon Press, London, 1968.

19. Lazan, B. J. and Goodman, L. E. "Material and InterfaceDamping (Section on 'Interface Shear Damping'). If Shockand Vibration Handbook. Ed. by C. Harris and C. Crede,Vol. II, Chap. 36, McGraw Hill, 1961.

53

AEDC-TR-76-41

20. Goodman, L. E. and Klumpp, J. H. "Analysis of SlipDamping with Reference to Turbine-Blade Vibration. "ASME Journal of Applied Mechanics. Vol. 23, September1956, pp. 421-429.

21. Lastman, G. J. "Solution of N Simultaneous First OrderDifferential Equations by Either the Runge-Kutta Methodor by the Adams-Moulton Method with a Runge-KuttaStarter, Using Partial Double Precision Arithmetic. "UTD2-03 -046, January 1964, University of Texas Com-putation Center (Program RKAM).

22. Alexander, W. K., Griffin, S. A., Holt, R. L., et al. "WindTunnel Model Parametric Study for Use in the Proposed8 ft x 10 ft High Reynolds Number Transonic Tunnel(HIRT) at Arnold Engineering Development Center. "AEDC-TR-73-47 (AD763725), March 1973.

23. Holt, R. L., Fatta, G. J., and Tyler, S. P. "Study ofModel Aeroelastic Characteristics in the Proposed HighReynolds Number Transonic Wind Tunnel (HIRT) inReference to the Aeroelastic Nature of the Flight Vehi-cle." AEDC-TR-75-62 (ADA016797), October 1975.

24. Pate, S. R. and Eaves, R. H., Jr. "Recent Advances in thePerformance and Testing Capabilities of the AEDC- VKFTunnel F (Hotshot) Hypersonic Facility." AIAA PaperNo. 74-84, January 30-February 1, 1974.

25. Hoerner, Sighard F. Fluid Dynamic Drag. Busteed, NewJersey, 1958.

54

AEDC-TR-76-41

NOMENCLATURE

Three basic axis systems are used in the sting planar motion anal-ysis of Section 2.5. Figure 6 illustrates these axis systems. andthey are further explained as follows:

This is a fixed inertial axis with XI alignedparallel to V00' The location of the origin isarbitrary.

This axis is fixed to the undeflected sting andmoves (rotates and translates) with the unde-fleeted sting. The origin is at the center ofthe sector rotation and XII is directed alongthe undeflected sting centerline. This is thebasic axis system from which sting deflectionsare measured.

This axis system is fixed to the i th lumped massof the system and rotates and translates withthis mass. XIII' is tangent to the deflectedsting centerline.l The origin is arbitrary butif chosen at the mass cg will eliminate termsinvolving ximi and zimi in the motion equations.

Additional symbols and nomenclature and abbreviations us ed hereinare defined as follows:

Instantaneous axial force on the model. lb.(Eq. (32) and Fig. 8)

Planform area of model, ft2

Planform area of the i th mass of the sting, ft2

Advanced transonic transport or advanced technol-ogy tran sport.

Initial value of qk at t = to (Eg. (39b))

Initial value of elk at t =to (Eg. (39b))Instantaneous axial-force coefficient (Eg. (32))

Angular deflection at i caused by unit force atj (Eg. (7))

55

AEDC·TR·76·41

CAM··IJ

C-S

CTF..IJ

CTM ..IJ

c

cg

DOF

Angular deflection at i caused by unit moment atj (Ego (7))

Constant defined by Ego (41)

Flexibility influence coefficients (Ego (6))

Static aerodynamic pitching-moment coefficient(Ego (28))

Aerodynamic pitch damping moment coefficientcaused by gR (Ego (27))

Total aerodynamic pitching-moment coefficient(Ego (27))

Slope of static pitching-moment coefficient.dCm/da (Ego (28))

Aerodynamic pitch damping moment coefficientcaused by a (Ego (27))Static aerodynamic normal-force coefficient(Ego (24))

Aerodynamic normal-force coefficient caused bygR (Ego (23))

Total aerodynamic normal-force coefficient(Ego (23))

Slope of static normal force coefficient.dCN / da (Ego (24))

Aerodynamic normal force coefficient causedby a (Eg. (23))Carbide-Steel composite sting

Translational deflection at i caused by unitforce at j (Eg. (7))

Translational deflection at i caused by unitmoment at j (Eg. (7))

Distance from beam neutral axis to surface.(Ego (2)). c =d/2 for circular cross-sectionstings

Center of gravity or. more correctly. center ofmass location

Degrees of freedom (number of independentcoordinates necessary to describe motion)

56

d

E

E1 to E5

F

FN

FCOFi.FCOMi

f k

g

I

ID

i. j. k

K

K ..1J

L

L1 to L5

LNcg

M

Mq

AEDC-TR-76-41

Aerodynamic reference length. ft or stingdiameter. in.

Modulus of elasticity or Young's Modulus.Ib/in. 2

E for beam sections in STING-1 computerprogram (Fig. 2)

Friction force (Eq. (33))

Static aerodynamic normal force. lb(FN = qO) S CN)

Structural damping constants (Eq. (34))

General functional form of C]k (Eq. (39a))

Gravitational constant. lbm/lbf /ft/ sec2

Area moment of inertia of beam cross section.in. 4 or ft4

Internal diameter of a hole through a stingsection. in. (Fig. 12)

Mass moment of inertia of the model or adiscrete lumped mass of the system represen-tation. ft-Ib-sec2 (Eq. (17d))

General indices

Linear spring constant. force/deflection

Stiffnes s influence coefficients (Eq. (Sa))

Length of sting. in. or ft

Length of individual sting sections (Fig. 2)

Length from model nose tip to cg position. in.(Figs. 12 and 17)

Total bending moment acting on sting atstation x, in. -lb, or ft-Ib

Total aerodynamic pitching moment, ft-Ib(Eq. (26))

(qO) S d2

/2 VO») Cm

q2

(q 0) S d / 2 V 00) Cm.

Q'

57

AEDC-TR-76-41

m

m·1m lli

m12i

m21·1

m22'1N

NT

Nl to NEND

NqN

bt

n

aDPM

QDF.1

QDMi

qoomaxRHl to RH5

Mass, Ib-sec2 /ft

Mass of i th system component, Ib-sec2 /ft

Quantity defined by Eqs. (36a) or (37a)

Quantity defined by Eqs. (36b) or (37b)

Quantity defined by Eqs. (36d) or (37d)

Quantity defined by Eqs. (36e) or (37e)

Number of DOF, N = 2n in the analysis ofSection 2. 5

Total aerodynamic normal force, lb (Eq. (22»

Station numbers in STING-l program (Fig. 2)

(qoo S d/2 V

(RHSF)i

(RHSM)i

R(Ni) -R(NEND)

S

T.1

t

u

VABS

W·1

WMWi to W5

w

x

AEDC-TR-76-41

Right-hand side of i th force Eq. (35a)(Eqs. (36c) or (37c))

Right-hand side of i th moment Eq. (35b)(Eqs. (36f) or (37f))

Radius at sting section discontinuities inSTING-i Program, in. (Fig. 2)

Reference area, ft2.qk' (Eq. (40a))

fk , (Eq. (40b))

Total kinetic energy of sting-model system,ft-1b (Section 2.5.2)

Kinetic energy of the i th mass of the systemrepresentation, ft-1b (Section 2. 5.2)

Time, sec

Initial time, usually taken as zero beforetransient conditions begin, sec

Total potential energy of the sting-balancesystem, ft-1b (Section 2.5.3)

Absolute velocity of a particle of mass (dm),ftl sec (Section 2.5.2)

Free-stream tunnel veo1city which is allowedto be a transient quantity, ftl sec. V (J) mustbe known to compute Qlplunge.

Weight of a concentrated mass, 1b

Weight of model, 1b

Specific weights of sting section material inSTING-i program, 1bl in. 3 (Fig. 2)

Specific weight of sting material, 1bl in. 3

General coordinate along undeflected stingcenterline measured from sector rotationpoint, in.; also general distance coordinate

Horizontal location of XII' ZII axis origin withrespect to the XI' ZI axis origin, ft

59

AEDC-TR-76-41

x·1

x·1

XIII·I

z

z·1

.z·1-z·1z·1

Horizontal velocity component of Xn ~ Znaxis (sting-model system moving as a rigidbody), ft/ sec

Horizontal acceleration component of Xn ~ Znaxis~ ft/ sec2

xII.~ location of i th mass measured along unde-1

flected sting centerline from model attachmentpoint~ in. (Eq. (16 c) ~ Fig. 6)

Location of cg of mass (mi) with respect toXIII. ~ ZnI. origin~ ft (Eq. (17b)~ Fig. 6c)

1 1

Horizontal location of particle of mass (dm) withrespect to XI ~ ZI axis origin~ ft (Eq. (14a) ~Fig. 6c)

xi ~ location of i th mass measured along unde-flected sting centerline along XII from XII,

Zn axis origin~ ft (Eq. (16c), Fig. 6)

Location of particle of mass (dm) with respectto Xnli ~ ZIni origin~ ft~ (Fig. 6b)

General deflection coordinate of sting center-line~ measured perpendicular to undeflectedsting centerline~ in.

Vertical location of XII ~ Zn axis origin withrespect to the XI ~ ZI axis origin~ ft

Vertical velocity Xn ~ Zn axis (sting-modelsystem moving as a rigid body) ~ ft/ sec

Vertical acceleration component of Xn ~ Znaxis~ ft/ sec2

ZII or qi ~ deflection of i th mass relative toundeflected sting centerline~ in. (Eq. (16a)~Fig. 6) also normalized deflections (eigenvectors)in Eq. (11)

dz i / dt or cli ~ ft/ sec

d2z i / dt2 or qi ~ ft/ sec

Location of cg of mass (mi) with respect toXln. ~ ZnI. axis origin~ ft (Eq. (17c)~ Fig. 6c)

1 1

60

Znose

zamplitude

e.1

,e.

1..e.

1

e.lav

e·Imax

AEDC-TR-76-41

Vertical location of particle of mass (dm) withrespect to XI' Zl axis origin, ft (Eq. (14b),Fig. 6c)

zi or qi' deflection of i th mass with respectto XII axis, in. (Eq. (16a), Fig. 6)

Location of particle mass (dm) with respectto XlIIi' ZIIli axis origin, ft (Fig. 6b)

Average amplitude of z, ft or in.

Maximum amplitude of zi' ft or in.

Amplitude of model nose tip translation, in.(Fig. 17)

Oscillation amplitude of z, ft or in. ,(z = Z - Z )amplitude max av

Pitch angle of sting-model acting as a rigidsystem (rotation of XII' ZII axis), deg or rad

Angular rate of XII' ZII axis rotation, rad/ sec

Angular acceleration of XII, ZII axis, rad/ sec 2

Angle of attack caused by transverse plungingvelocity components, deg or rad (Eqs. (29), (43),and (44»

Total angle of attack, deg or rad (Eq. (29»

daTI dt, rad/ sec (Eq. (30»

Moment arm of axial force, ft (£:'i = xiei -zl + zi) (Fig. 7)

Maximum bending strain at station x, in/ in. ,(Eq. (3»

qi+n' rotational deflection of sting at station xi'deg or rad (Eq. (16b»,qi+n' rad/ sec

'q' rad/ sec2i+n'Average amplitude of ei , deg

Maximum amplitude of ei , deg

61

AEDC-TR-76-41

(J.lamplitude Oscillation amplitude of (Ji. deg ((Ji l't d =(J. _ (J. ) amp 1 u e

lmax lav

1/w2, (CTFij mi) matrix eigenvalue, (Eq. (11))

Maximum bending stress at station x. lb/ in. 2(Eq. (2))

Material yield stress. lb/ in. 2 (Table 1)

Frequency of vibration of the i th mode. cps(i = 1.2.3.etc.)

62